The power rule is a fantastic "shortcut" for finding the derivatives of basic polynomials. Between the power rule and the basic definition of the derivative of a constant, a great number of polynomial derivatives can be identified with little effort  often in your head!
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 Khan Academy: Calculus: Derivatives 3
Guidance
In this lesson, we will develop formulas and theorems that will calculate derivatives in more efficient and quick ways. Look for these theorems in boxes throughout the lesson.
The Derivative of a Constant
Theorem: If where c is a constant, then Proof: 

Theorem: If is a constant and is differentiable at all , then . In simpler notation 

The Power Rule
Theorem: (The Power Rule)
If
n
is a positive integer, then for all real values of
x


Example A
Find for
Solution
 If for all , then for all
 We can also write
Example B
Find the derivative of
Solution
 ..... Restate the function
 ..... Apply the Commutative Law
 ..... Apply the Power Rule
 ..... Simplify
Example C
Find the derivative of
Solution
 ..... Restate
 ..... Rules of exponents
 ..... By the Commutative law
 ..... Apply the Power Rule
 ..... Simplify
 ..... Simplify again
 ..... Use rules of exponents
Vocabulary
A theorem is a statement accepted to be true based on a series of reasoned statements already accepted to be true. In the context of this lesson, a theorem is a rule that allows a quick calculation of the derivative of functions of different types.
A proof is a series of true statements leading to the acceptance of truth of a more complex statement.
Guided Practice
Questions
Find the derivatives of:
1)
2)
3)
4)
Solutions
1) By the power rule:
 If then
2) Special application of the power rule:
3) Restate the function:
 Using rules of exponents (from Algebra):
 Apply the Power Rule:
 Simplify:
 Rules of exponents:
 Simplify:
4) Restate the function:
 Rules of exponents:
 Power Rule:
 Simplify:
 Rules of exponents:
Practice
 State the Power Rule.
Find the derivative:
 given when
 given what is
 when
 given what is